## Nov 12, 2009

### 3 Families

Three families make a remarkable discovery. The sum of the ages of their members are all the same, the sum of the squares of the ages of their members are all the same, and the sum of the cubes of the ages of their members are all the same. Everyone in all 3 families has a different age, and nobody is more than 100 years old.

What is the smallest possible sum of their ages? Can this be done with 4 families?

1. What is the number of memebers in each family ? Is it required ?

2. I haven't been able to solve the problem yet...I had assumed it to be unknown since it wasn't mentioned in the problem when i first read it :)

3. assuming 2 persons each in a family,

a+b=c+d=e+f=s
a^2+b^2=c^2+d^2=....
similarly sum of cubes same

all satisfied together if,
ab=cd=ef;
a(s-a)=c(s-c)=e(s-e)
so, a+c=s=e+c
so,b=c=f not possible for distinct ages.

so no solution.

4. AnonymousJune 24, 2010

@Anonymous. But what if there are not just 2 persons in the family?

5. Tanishq GoyalOctober 08, 2010

yes..second anonymous is right!!
may be this can be solved for just one particular value of number of members in each family..and more over..it is possible that each family has different number of members..there is no restriction on that also!!